TRANSFORMATIONS OF RANDOM VARIABLES 1.1. Definition. We ...

TRANSFORMATIONS OF RANDOM VARIABLES

1. I NTRODUCTION 1.1. Definition. We are often interested in the probability distributions or densities of functions of one or more random variables. Suppose we have a set of random variables, X1, X2 , X3 , . . . Xn , with a known joint probability and/or density function. We may want to know the distribution of some function of these random variables Y = φ(X1, X2, X3, . . . Xn). Realized values of y will be related to realized values of the X’s as follows y = Φ (x1, x2, x3 , · · · , xn)

(1)

A simple example might be a single random variable x with transformation y = Φ (x) = log (x)

(2)

1.2. Techniques for finding the distribution of a transformation of random variables. 1.2.1. Distribution function technique. We find the region in x1 , x2 , x3 , . . . xn space such that Φ(x1 , x2 , . . . xn ) ≤ φ. We can then find the probability that Φ(x1 , x2 , . . . xn ) ≤ φ, i.e., P[ Φ(x1 , x2 , . . . xn ) ≤ φ ] by integrating the density function f(x1, x2 , . . . xn ) over this region. Of course, FΦ (φ) is just P[ Φ ≤ φ ]. Once we have FΦ (φ), we can find the density by integration. 1.2.2. Method of transformations (inverse mappings). Suppose we know the density function of x. Also suppose that the function y = Φ (x) is differentiable and monotonic for values within its range for which the density f(x) 6=0. This means that we can solve the equation y = Φ (x) for x as a function of y. We can then use this inverse mapping to find the density function of y. We can do a similar thing when there is more than one variable X and then there is more than one mapping Φ. 1.2.3. Method of moment generating functions. There is a theorem (Casella [2, p. 65] ) stating that if two random variables have identical moment generating functions, then they possess the same probability distribution. The procedure is to find the moment generating function for Φ and then compare it to any and all known ones to see if there is a match. This is most commonly done to see if a distribution approaches the normal distribution as the sample size goes to infinity. The theorem is presented here for completeness. Theorem 1. Let FX (x) and FY (y) be two cummulative distribution functions all of whose moments exist. Then a: If X and Y hae bounded support, then FX (u) and FY (u) for all u if an donly if E Xr = E Yr for all integers r = 0,1,2, . . . . b: If the moment generating functions exist and MX (t) = MY (t) for all t in some neighborhood of 0, then FX (u) = FY (u) for all u. For further discussion, see Billingsley [1, ch. 21-22] . Date: August 9, 2004. 1

2

TRANSFORMATIONS OF RANDOM VARIABLES

2. D ISTRIBUTION F UNCTION T ECHNIQUE 2.1. Procedure for using the Distribution Function Technique. As stated earlier, we find the region in the x1 , x2 , x3 , . . . xn space such that Φ(x1 , x2 , . . . xn ) ≤ φ. We can then find the probability that Φ(x1 , x2 , . . . xn ) ≤ φ, i.e., P[ Φ(x1, x2 , . . . xn ) ≤ φ ] by integrating the density function f(x1, x2 , . . . xn ) over this region. Of course, FΦ (φ) is just P[ Φ ≤ φ ]. Once we have FΦ(φ), we can find the density by integration. 2.2. Example 1. Let the probability density function of X be given by f (x) =



6 x (1 − x), 0 < x < 1 0 otherwise

(3)

Now find the probability density of Y = X3 . Let G(y) denote the value of the distribution function of Y at y and write G( y ) = P ( Y ≤ y ) = P ( X3 ≤ y )   = P X ≤ y1/3 Z y1/3 = 6x (1 − x)dx =

Z

(4)

0 y 1/3


6 x − 6 x2 d x

0


1/3 3 x2 − 2 x3 |y0

=

= 3 y2/3 − 2y Now differentiate G(y) to obtain the density function g(y) d G (y) dy  d  2/3 = − 2y 3y dy

g(y) =

= 2 y− 1/3 − 2 = 2(y

−1/3

(5)

− 1), 0 < y < 1

2.3. Example 2. Let the probability density function of x1 and of x2 be given by f ( x 1 , x2 ) =



2 e− x1 − 2 x2 , x1 > 0 , x2 > 0 0 otherwise

(6)

Now find the probability density of Y = X1 + X2. Given that Y is a linear function of X1 and X2, we can easily find F(y) as follows. Let FY (y) denote the value of the distribution function of Y at y and write

TRANSFORMATIONS OF RANDOM VARIABLES

FY (y) = P (Y ≤ y) Z y Z y − x2 = 2 e− x1 − 2 x2 d x1 d x2 Z0 y 0 = − 2 e− x1 − 2 x2 |y0 − x2 d x2 0 Z y 

 = − 2 e− y + x2 − 2 x2 − −2 e− 2 x2 d x2 Z0 y = − 2 e− y − x2 + 2 e− 2 x2 d x2 0 Z y = 2 e− 2 x2 − 2 e− y − x2 d x2

3

(7)

0

Now integrate with respect to x2 as follows F (y) = P (Y ≤ y) Z y = 2 e− 2 x2 − 2 e− y − x2 d x2 0

= − e− 2 x2 + 2 e− y − x2 |y0   = − e− 2 y + 2 e− y − y − − e0 + 2 e− y

(8)

= e− 2 y − 2 e− y + 1 Now differentiate FY (y) to obtain the density function f(y) d F (y) dy
d = e− 2 y − 2 e− y + 1 dy = − 2 e− 2 y + 2 e− y

FY (y) =

(9)

= 2 e− 2 y (− 1 + e y ) 2.4. Example 3. Let the probability density function of X be given by 2 −1 x − µ 1 √ · e 2 ( σ ) −∞ < x < ∞ σ 2π   1 ( x − µ )2 = √ −∞ < x < ∞ · exp − 2 σ2 2 π σ2

fX ( x ) =

(10)

Now let Y = Φ(X) = eX . We can then find the distribution of Y by integrating the density function of X over the appropriate area that is defined as a function of y. Let FY (y) denote the value of the distribution function of Y at y and write

4

TRANSFORMATIONS OF RANDOM VARIABLES

FY (y) = P ( Y ≤ y) eX ≤ y

=P Z =




= P ( X ≤ ln y) , y > 0   1 ( x − µ )2 √ d x, y > 0 · exp − 2 σ2 2 π σ2

ln y −∞

(11)

Now differentiate FY (y) to obtain the density function f(y). In this case we will need the rules for differentiating under the integral sign. They are given by theorem 2 which we state below without proof. Theorem 2. Suppose that f and

∂f ∂x

are continuous in the rectangle

R = { (x, t) : a ≤ x ≤ b , c ≤ t ≤ d} and suppose that u0 (x) and u1 (x) are continuously differentiable for a ≤ x ≤ b with the range of u0 (x) and u1 (x) in (c, d). If ψ is given by ψ (x) =

Z

u1 (x)

(12)

f(x, t) d t u0 (x)

then dψ ∂ = dx ∂x

Z

u1 (x)

(13)

f(x, t) d t u0 (x)

= f ( x, u1 (x) )

d u1 (x) d u0 − f (x, u0 (x)) + dx dx

Z

u1 (x) u0 (x)

∂ f (x, t) dt ∂x

If one of the bounds of integration does not depend on x, then the term involving its derivative will be zero. For a proof of theorem 2 see (Protter [3, p. 425] ). Applying this to equation 11 we obtain i h 2 R ln y FY (y) = − ∞ √ 1 2 · exp − ( x2−σ2µ ) d x , y > 0 2 π σ i   h ( ln y − µ )2 1 √ 1 FY0 (y) = fY (y) = · exp − 2 2σ y 2 π σ2 + =



R ln

y −∞

d dy

√ 1 y 2 π σ2



√ 1 2 π σ2

· exp

h

−

· exp

h

−

( ln y − µ )2 2 σ2

( x − µ )2 2 σ2

i

i

dx

(14)

TRANSFORMATIONS OF RANDOM VARIABLES

5

TABLE 1. Outcomes, Probabilities and Number of Heads from Tossing a Coin Four Times. Element of sample space Probability HHHH 81/625 HHHT 54/625 HHTH 54/625 HTHH 54/625 THHH 54/625 HHTT 36/625 HTHT 36/625 HTTH 36/625 THHT 36/625 THTH 36/625 TTHH 36/625 HTTT 24/625 THTT 24/625 TTHT 24/625 TTTH 24/625 TTTT 16/625 3. M ETHOD

Value of random variable X (x) 4 3 3 3 3 2 2 2 2 2 2 1 1 1 1 0

OF TRANSFORMATIONS ( SINGLE VARIABLE )

3.1. Discrete examples of the method of transformations. 3.1.1. One-to-one function. Find a formula for the probability distribution of the total number of heads obtained in four tosses of a coin where the probability of a head is 0.60. The sample space, probabilities and the value of the random variable are given in table 1. From the table we can determine the probabilities as 16 96 216 216 81 , P (X = 1) = , P (X = 2) = , P (X = 3) = , P (X = 4) = 625 625 625 625 625 The probability of 3 heads and one tail for all possible combinations is

P (X = 0) =

        3 3 3 2 5 5 5 5 or  3  1 3 2 . 5 5 Similarly the probability of one head and three tails for all possible combinations is         3 2 2 2 5 5 5 5 or

6

TRANSFORMATIONS OF RANDOM VARIABLES

 1  3 3 2 5 5 There is one way to obtain four heads, four ways to obtain three heads, six ways to obtain two heads, four ways to obtain one head and one way to obtain zero heads. These five numbers are 1, 4, 6, 4, 1 which is a set of binomial coefficients. We can then write the probability mass function as f(x) =



4 x

  x  4 − x 3 2 for x = 0, 1, 2, 3, 4 5 5

(15)

This, of course, is the binomial distribution. The probabilities of the various possible random variables are contained in table 2. TABLE 2. Probability of Number of Heads from Tossing a Coin Four Times Number of Heads x 0 1 2 3 4

Probability f(x) 16/625 96/625 216/625 216/625 81/625

Now consider a transformation of X in the form Y = 2X2 + X. There are five possible outcomes for Y, i.e., 0, 3, 10, 21, 36. Given that the function is one-to-one, we can make up a table describing the probability distribution for Y. TABLE 3. Probability of a Function of the Number of Heads from Tossing a Coin Four Times. Y = 2 * (# heads)2 + # of heads Probability Number of Heads f(x) x y g(y) 0 16/625 0 16/625 1 96/625 3 96/625 2 216/625 10 216/625 3 216/625 21 216/625 4 81/625 36 81/625

3.1.2. Case where the transformation is not one-to-one. Now let the transformation of X be given by Z = (6 - 2X)2. The possible values for Z are 0, 4, 16, 36. When X = 2 and when X = 4, Y = 4. We can find the probability of Z by adding the probabilities for cases when X gives more than one value as shown in table 4.

TRANSFORMATIONS OF RANDOM VARIABLES

7

TABLE 4. Probability of a Function of the Number of Heads from Tossing a Coin Four Times (not one-to-one). Y = (6 - (# heads))2 Number of Heads x y g(y) 0 3 216/625 4 2, 4 216/625 + 81/625 = 297 16 1 96/625 36 0 16/625 3.2. Intuitive Idea of the Method of Transformations. The idea of a transformation is to consider the function that maps the random variable X into the random variable Y. The idea is that if we can determine the values of X that lead to any particular value of Y, we can obtain the probability of Y by summing the probabilities of those values of X that mapped into Y. In the continuous case, to find the distribution function, we want to integrate the density of X over the portion of its space that is mapped into the portion of Y in which we are interested. Suppose for example that both X and Y are defined on the real line with 0 ≤ X ≤ 1 and 0 ≤ Y ≤ 10. If we want to know G(5), we need to integrate the density of X over all values of x leading to a value of y less than five, where G(y) is the probability that Y is less than five. 3.3. General formula when the random variable is discrete. Consider a transformation defined by y = Φ(x). The function Φ defines a mapping from the sample space of the variable X, to a sample space for the random variable Y. If X is discrete with frequency function pX , then Φ(X) is discrete and has frequency function p Φ(X) (t) =

X

pX (x)

(16)

x : Φ (x) = t

=

X

pX (x)

x  Φ− 1 (t)

The process is simple in this case. One identifies g−1(t) for each t in the sample space of the random variable Y, and then sums the probabilities. 3.4. General change of variable or transformation formula. Theorem 3. Let fX (x) be the value of the probability density of the continuous random variable X at x. If the function y = Φ(x) is differentiable and either increasing or decreasing (monotonic) for all values within the range of X for which fX (x) 6= 0, then for these values of x, the equation y = Φ(x) can be uniquely solved for x to give x = Φ−1(y) = w(y) where w(·) = Φ−1(·). Then for the corresponding values of y, the probability density of Y =Φ(X) is given by   −1  d Φ −1 (y)  Φ ( y ) · f  X  d y   d w (y) d Φ ( x ) fX [ w (y) ] · d y 6= 0 g (y) = fY (y) = (17) dx  0  fX [ w (y) ] · | w (y) |    0 otherwise

8

TRANSFORMATIONS OF RANDOM VARIABLES

Proof. Consider the digram in figure 1.

F IGURE 1. y = Φ(x) is an increasing function.

As can be seen from in figure 1, each point on the y axis maps into a point on the x axis, that is, X must take on a value between Φ−1 (a) and Φ−1 (b) when Y takes on a value between a and b. Therefore
P (a < Y < b) = P Φ −1 (a) < X < Φ −1 (b) R Φ −1 (b) (18) = Φ −1 (a) fX (x) d x What we would like to do is replace x in the second line with y, and Φ−1 (a) and Φ−1 (b) with a and b. To do so we need to make a change of variable. Consider how we make a u substitution when we perform integration or use the chain rule for differentiation. For example if u = h(x) then du = h0 (x) dx. So if x = Φ−1(y), then dx = Then we can write

d Φ −1 (y ) dy. dy

TRANSFORMATIONS OF RANDOM VARIABLES

Z

fX (x) d x =

Z

Φ −1 (y)

fX

9


d Φ −1(y) dy dy

(19)

For the case of a definite integral the following lemma applies. Lemma 1. If the function u = h(x) has a continuous derivative on the closed interval [a, b] and f is continuous on the range of h, then Z

b 0

f ( h (x)) h (x) d x = a

Z

h (b)

(20)

f (u) du h (a)

Using this lemma or the intuition from equation 19 we can then rewrite equation 18 as follows
P ( a < Y < b ) = P Φ −1 (a) < X < Φ −1 (b) R Φ −1 ( b ) = Φ −1 (a) fX (x) d x
Rb −1 = a fX Φ −1 ( y ) d Φ d y (y ) d y

(21)

The probability density function, fY (y), of a continuous random variable Y is the function f(·) that satisfies Z

P (a < Y < b) = F (b) − F (a) =

b

fY (t) d t

(22)

a

This then implies that the integrand in equation 21 is the density of Y, i.e., g(y), so we obtain g (y) = fY (y) = fX



Φ− 1 (y)



·

d Φ −1 (y) dy

(23)

as long as d Φ −1 (y) dy exists. This proves the lemma if Φ is an increasing function. Now consider the case where Φ is a decreasing function as in figure 2. As can be seen from figure 2, each point on the y axis maps into a point on the x axis, that is, X must take on a value between Φ−1 (a) and Φ−1(b) when Y takes on a value between a and b. Therefore P (a < Y < b) = P Z =

Φ −1 (b) < X < Φ −1 (a) Φ −1 ( a )

fX (x) d x Φ −1 ( b )

Making a change of variable for x = Φ−1(y) as before we can write




(24)

10

TRANSFORMATIONS OF RANDOM VARIABLES

F IGURE 2. y = Φ(x) is a decreasing function.

Φ −1 (b) < X < Φ −1 (a)

P (a < Y < b) = P Z = =




Φ −1 (a)

fX (x) d x

Φ −1 (b) Z a

fX

b

= −

Z

b

fX a


d Φ −1 (y) dy dy
d Φ −1 (y) Φ −1 (y) dy dy

Φ −1 (y)

Because d Φ −1 (y) dx 1 = = dy dy dy dx when the function y = Φ(x) is increasing and

(25)

TRANSFORMATIONS OF RANDOM VARIABLES

11

d Φ −1 (y) dy

−

is positive when y = Φ(x) is decreasing, we can combine the two cases by writing g (y) = fY (y) = fX



Φ− 1 (y)



·

d Φ −1 (y ) dy



(26) 

3.5. Examples. 3.5.1. Example 1. Let X have the probability density function given by fX (x) =



1 2

x, 0 ≤ x ≤ 2 0 elsewhere

(27)

Find the density function of Y = Φ(X) = 6X - 3. Notice that fX (x) is positive for all x such that 0 ≤ x ≤ 1. The function Φ is increasing for all X. We can then find the inverse function Φ−1 as follows (28)

y = 6x − 3 ⇒ 6x = y + 3 y + 3 ⇒ x = = Φ −1 (y) 6 We can then find the derivative of Φ−1 with respect to y as d Φ −1 d = dy dy 1 = 6



y + 3 6



(29)

The density of y is then 

−1

g (y) = fY (y) = fX Φ (y)     1 3 + y 1 = 6 2 6



d Φ −1 (y) · dy 3 + , 0 ≤ 6



y

(30) ≤ 2

For all other values of y, g(y) = 0. Simplifying the density and the bounds we obtain g (y) = fY (y) =



3+y 72

, −3 ≤ y ≤ 9 0 elsewhere

(31)

12

TRANSFORMATIONS OF RANDOM VARIABLES

3.5.2. Example 2. Let X have the probability density function given by fX (x). Then consider the transformation Y = Φ(X) = σX + µ, σ 6= 0. The function Φ is increasing for all X. We can then find the inverse function Φ−1 as follows (32)

y = σx + µ ⇒ σx =y − µ y − µ = Φ −1 (y) ⇒ x = σ We can then find the derivative of Φ−1 with respect to y as d Φ −1 d = dy dy 1 = σ



y − µ σ



(33)

The density of y is then 

 Φ− 1 (y)   y−µ = fX · σ

fY (y) = fX

d Φ −1 (y ) · dy 1 σ



3.5.3. Example 3. Let X have the probability density function given by  e −x , 0 ≤ x ≤ ∞ fX (x) = 0 elsewhere

(34)

(35)

Find the density function of Y = X1/2. Notice that fX (x) is positive for all x such that 0 ≤ x ≤ ∞. The function Φ is increasing for all X. We can then find the inverse function Φ−1 as follows 1

y = x2 ⇒ y2 = x ⇒ x = Φ −1 ( y ) = y2

(36)

We can then find the derivative of Φ−1 with respect to y as d Φ −1 dy

= ddy y2 = 2y

(37)

The density of y is then fY (y) = fX = e− y

 2

Φ− 1 ( y ) |2y|



·

d Φ −1 (y ) dy

A graph of the two density functions is shown in figure 3 .



(38)

TRANSFORMATIONS OF RANDOM VARIABLES

13

F IGURE 3. The Two Density Functions.

1

Value of Density Function

0.8

fHxL

0.6

0.4

gHyL

0.2

1 4. M ETHOD

2

3

4

OF TRANSFORMATIONS ( MULTIPLE VARIABLES )

4.1. General definition of a transformation. Let Φ be any function from Rk to Rm , k, m ≥ 1, such that Φ−1 (A) = x  Rk : Φ(x) A  ßk for every A  ßm where ßm is the smallest σ - field having all the open rectangles in Rm as members. If we write y = Φ(x), the function Φ defines a mapping from the sample space of the variable X (Ξ) to a sample space (Y) of the random variable Ψ. Specifically Φ(x) : Ξ → Ψ

(39)

Φ−1 (A) = { x  Ψ : Φ (x)  A }

(40)

and

4.2. Transformations involving multiple functions of multiple random variables. Theorem 4. Let fX1 X2 (x1 x2 ) be the value of the joint probability density of the continuous random variables X1 and X2 at (x1, x2). If the functions given by y1 = u1 (x1, x2) and y2 = u2 (x1, x2 ) are partially differentiable with respect to x1 and x2 and represent a one-to-one transformation for all values within the range of X1 and X2 for which fX1 X2 (x1 x2) 6= 0 , then, for these values of x1 and x2 , the equations y1 = u1 (x1, x2) and y2 = u2 (x1, x2) can be uniquely solved for x1 and x2 to give x1 = w1 (y1 , y2 ) and x2 = w2 (y1 , y2 ) and for corresponding values of y1 and y2, the joint probability density of Y1 = u1(X1 , X2 ) and Y2 = u2(X1 , X2) is given by fY1 Y2 ( y1 y2 ) = fX1 X2 [ w1 ( y1 , y2 ) , w2 ( y1 y2 ) ] · | J | where J is the Jacobian of the transformation and is defined as the determinant

(41)

14

TRANSFORMATIONS OF RANDOM VARIABLES

J =

∂x1 y1 ∂x2 y1



∂x1 y2 ∂x2 y2

(42)

At all other points fY1 Y2 (y1 y2 ) = 0 . 4.3. Example. Let the probability density function of X1 and X2 be given by

fX1 X2 (x1, x2) =

(

e − ( x1 + x2 ) 0

for x1 ≥ 0, x2 ≥ 0 elsewhere

Consider two random variables Y1 and Y2 be defined in the following manner. Y 1 = X1 + X2 X1 Y2 = X1 + X2

(43)

To find the joint density of Y1 and Y2 and also the marginal density of Y2 . We first need to solve the system of equations in equation 43 for X1 and X2. Y1 Y2 ⇒ X1 ⇒ Y2 ⇒ Y2 ⇒ Y1 Y2 ⇒ X2 ⇒ X1

= X1 + X2 = X1 X+1 X2 = Y 1 − X2 = Y1 Y−1 X−2 X+2 X2 = Y1 Y−1 X2 = Y 1 − X2 = Y1 − Y1 Y2 = Y1 (1 − Y2 ) = Y1 − ( Y1 − Y1 Y2 ) = Y1 Y2

(44)

The Jacobian is given by

J =

∂x1 y1 ∂x2 y1

∂x1 y2 ∂x2 y2



y2 y1 = 1 − y2 − y1 = − y2 y1 − y1 ( 1 − y2 ) = − y2 y1 − y1 1 + y1 y2 = − y1

(45)

This transformation is one-to-one and maps the domain of X (Ξ) given by x1 > 0 and x2 > 0 in the x1 x2 -plane into the domain of Y(Ψ) in the y1 y2-plane given by y1 > 0 and 0 < y2 < 1. If we apply theorem 4 we obtain

TRANSFORMATIONS OF RANDOM VARIABLES

fY1

Y2

(y1 y2 ) = fX1 X2 [ w1 ( y1 , y2 ) , w2 ( y1 y2 ) ] · | J | = e − ( y1 y2 + y1 − y1 y2 ) | − y1 | = e − y1 | − y1 | = y1 e − y1

15

(46)

Considering all possible values of values of y1 and y2 we obtain ( y1 e − y1 for x1 ≥ 0 , 0 < x2 < 1 fY1 Y2 (y1 , y2 ) = 0 elsewhere We can then find the marginal density of Y2 by integrating over y1 as follows R∞ fY2 (y1 , y2 ) = R0 fY2 Y2 (y1 , y2 ) y1 ∞ = 0 y1 e− y1 d y1

(47)

We make a uv substitution to integrate where u, v, du, and dv are define as u = y1 v = − e− y1 d u = d y1 dv = e− y1 d y1

(48)

R∞ fY2 (y1 , y2 ) = 0 y1 e− y1 d y1 R ∞ = − y1 e− y1 |∞ − e− y1 d y1 0 − 0 − y1 = (0 − 0 ) − (e ) |∞ 0 − y1 ∞ = 0 − (e ) |0
= 0 − e− ∞ − e0 =0 − 0 + 1 =1

(49)

This then implies

for all y2 such that 0 < y2 < 1.

16

TRANSFORMATIONS OF RANDOM VARIABLES

A graph of the joint densities and the marginal density follows. The joint density of (X1, X2) is shown in figure 4. F IGURE 4. Joint Density of X1 and X2 .

x1

2

1

0

3 4 0.2 0.15 0.1

f12Hx1,x2L

0.05 0

1

2

3

0 4

x2 This joint density of (Y1, Y2) is contained in figure 5. F IGURE 5. Joint Density of Y1 and Y2.

2

0

4

y1 6 8

0.3 0.2 f Hy ,y L 1 2 0.1

0

0.25

0.5 y2

0.75

0 1

TRANSFORMATIONS OF RANDOM VARIABLES

17

This marginal density of Y2 is shown graphically in figure 6. F IGURE 6. Marginal Density of Y2.

4

y1

2

0

6 8

2 1.5 1 0.5

0

0.25

0.5 y2

0.75

0 1

f2Hy2L

18

TRANSFORMATIONS OF RANDOM VARIABLES

R EFERENCES [1] Billingsley, P. Probability and Measure. 3rd edition. New York: Wiley, 1995 [2] Casella, G. and R.L. Berger. Statistical Inference. Pacific Grove, CA: Duxbury, 2002 [3] Protter, Murray H. and Charles B. Morrey, Jr. Intermediate Calculus. New York: Springer-Verlag, 1985